This site talks about growing watermelons.
This site has brix refractometers for sale. They are used to measure the sugar content of watermelons though I am not sure how to use one.
A maker of refractometers can be found at this site:
I am also looking into what some brown spots that appeared on several watermelon plants are. I wonder if I should remove the leaves that are affected or let the plant take care of them. It appears to be on 3 of maybe 21 plants. The spots may be on only one variety of watermelon. According to the University of Georgia site (http://www.uga.edu/vegetable/watermelon.html) it could be Anthracnose. Reading the details of this Colletotrichum lagenarium fungus this does not sound good. It is on the oldest leaves at the center of the plants.
Of course, it might be a Cercospora citrullina fungus that can cause leaf spots described at this Texas A&M site (http://aggie-horticulture.tamu.edu/southtex/info/Watermelon.html). It does not appear as harmful but the pictures at the Univ. of Georgia site look closer to what I am seeing. Here is a list of watermelon problems http://plantpathology.tamu.edu/Texlab/Vegetables/wmelon/wmmenu.htm
I think I am going to remove dying leaves and spray a mixture of baking soda/vegetable oil on leaves that have few spots and those near by. I might try sulfur and hydrogen peroxide as well. Fingers crossed this will not spread with all the overcast and wet weather that is occurring.